Low density parity check encoder having length of 64800 and code rate of 5/15, and low density parity check encoding method using the same

ABSTRACT

A low density parity check (LDPC) encoder, an LDPC decoder, and an LDPC encoding method are disclosed. The LDPC encoder includes first memory, second memory, and a processor. The first memory stores an LDPC codeword having a length of 64800 and a code rate of 5/15. The second memory is initialized to 0. The processor generates the LDPC codeword corresponding to information bits by performing accumulation with respect to the second memory using a sequence corresponding to a parity check matrix (PCM).

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.15/907,025 filed Feb. 27, 2018, now U.S. Pat. No. 10,432,226 issued Oct.1, 2019, which is a continuation of U.S. patent application Ser. No.15/642,199 filed Jul. 5, 2017, now U.S. Pat. No. 9,941,902 issued Apr.10, 2018, which is a continuation of U.S. patent application Ser. No.15/271,941 filed Sep. 21, 2016, now U.S. Pat. No. 9,735,808 issued Aug.15, 2017, which is a continuation of and claims priority to U.S.application Ser. No. 14/495,803 filed Sep. 24, 2014, now U.S. Pat. No.9,496,896 issued Nov. 15, 2016, which claims the benefit of KoreanPatent Application Nos. 10-2014-0106181 and 10-2014-0117504, filed Aug.14, 2014 and Sep. 4, 2014, respectively, which are hereby incorporatedby reference herein in their entirety.

BACKGROUND 1. Technical Field

The present disclosure relates generally to a low density parity check(LDPC) code that is used to correct errors occurring over a wirelesschannel, and, more particularly, to an LDPC code that is applicable to adigital broadcasting system.

2. Description of the Related Art

Current terrestrial television (TV) broadcasting generates co-channelinterference across an area within a distance that is three times aservice radius, and thus the same frequency cannot be reused in the areawithin the distance that is three times the service radius. An area inwhich the same frequency cannot be reused is called a white space.Spectrum efficiency significantly deteriorates due to the occurrence ofa white space.

Accordingly, there arises a need for the development of a transmissiontechnology that facilitates the elimination of a white space and thereuse of a frequency with an emphasis on reception robustness in orderto improve spectrum efficiency.

In response to this, the paper “Cloud Transmission: A New Spectrum-ReuseFriendly Digital Terrestrial Broadcasting Transmission System” publishedon September of 2012 in IEEE Transactions on Broadcasting, Vol. 58, No.3 proposes a terrestrial cloud transmission technology that facilitatesreuse, does not generate a white space, and makes the construction andoperation of a single frequency network easy.

Using this terrestrial cloud transmission technology, a broadcastingstation can transmit the same nationwide content or locally differentcontent over a single broadcasting channel. However, for this purpose, areceiver should receive one or more terrestrial cloud broadcast signalsin an area in which signals transmitted from different transmittersoverlap each other, that is, an overlap area, over a single frequencynetwork, and then should distinguish and demodulate the receivedterrestrial cloud broadcast signals. That is, the receiver shoulddemodulate one or more cloud broadcast signals in a situation in whichco-channel interference is present and the timing and frequencysynchronization between transmitted signals are not guaranteed.

Meanwhile, Korean Patent Application Publication No. 2013-0135746entitled “Low Density Parity Check Code for Terrestrial CloudTransmission” discloses an LDPC code that is optimized for terrestrialcloud transmission and exhibits excellent performance at low code rate(<0.5).

However, Korean Patent Application Publication No. 2013-0135746 isdirected to a code length completely different from an LDPC code lengthused in the DVB broadcast standard, etc., and does not teach a specificLDPC encoding method.

SUMMARY

At least one embodiment of the present invention is directed to theprovision of a new LDPC codeword having a length of 64800 and a coderate of 5/15, which is capable of being used for general purposes.

At least one embodiment of the present invention is directed to theprovision of an LDPC encoding technique that is capable of efficientlyperforming LDPC encoding using a sequence having a number of rows equalto a value that is obtained by dividing the sum of the length of thesystematic part of an LDPC codeword, that is, 21600, and the length ofthe first parity part of the LDPC codeword, that is, 1440, by 360.

In accordance with an aspect of the present invention, there is providedan LDPC encoder, including first memory configured to store an LDPCcodeword having a length of 64800 and a code rate of 5/15; second memoryconfigured to be initialized to 0; and a processor configured togenerate the LDPC codeword corresponding to information bits byperforming accumulation with respect to the second memory using asequence corresponding to a parity check matrix (PCM).

The accumulation may be performed at parity bit addresses that areupdated using the sequence corresponding to the PCM.

The LDPC codeword may include a systematic part corresponding to theinformation bits and having a length of 21600, a first parity partcorresponding to a dual diagonal matrix included in the PCM and having alength of 1440, and a second parity part corresponding to an identitymatrix included in the PCM and having a length of 41760.

The sequence may have a number of rows equal to the sum of a valueobtained by dividing a length of the systematic part, that is, 21600, bya circulant permutation matrix (CPM) size corresponding to the PCM, thatis, 360, and a value obtained by dividing a length of the first paritypart, that is, 1440, by the CPM size.

The sequence may be represented by the following Sequence Table:

Sequence Table 1st row: 221 1011 1218 4299 7143 8728 11072 15533 1735633909 36833 2nd row: 360 1210 1375 2313 3493 16822 21373 23588 2365626267 34098 3rd row: 544 1347 1433 2457 9186 10945 13583 14858 1919534606 37441 4th row: 37 596 715 4134 8091 12106 24307 24658 34108 4059142883 5th row: 235 398 1204 2075 6742 11670 13512 23231 24784 2791534752 6th row: 204 873 890 13550 16570 19774 34012 35249 37655 3988542890 7th row: 221 371 514 11984 14972 15690 28827 29069 30531 3101843121 8th row: 280 549 1435 1889 3310 10234 11575 15243 20748 3046936005 9th row: 223 666 1248 13304 14433 14732 18943 21248 23127 3852939272 10th row: 370 819 1065 9461 10319 25294 31958 33542 37458 3968140039 11st row: 585 870 1028 5087 5216 12228 16216 16381 16937 2713227893 12nd row: 164 167 1210 7386 11151 20413 22713 23134 24188 3677138992 13rd row: 298 511 809 4620 7347 8873 19602 24162 29198 34304 4114514th row: 105 830 1212 2415 14759 15440 16361 16748 22123 32684 4257515th row: 659 665 668 6458 22130 25972 30697 31074 32048 36078 3712916th row: 91 808 953 8015 8988 13492 13987 15979 28355 34509 39698 17throw: 594 983 1265 3028 4029 9366 11069 11512 27066 40939 41639 18th row:506 740 1321 1484 10747 16376 17384 20285 31502 38925 42606 19th row:338 356 975 2022 3578 18689 18772 19826 22914 24733 27431 20th row: 7091264 1366 4617 8893 25226 27800 29080 30277 37781 39644 21st row: 8401179 1338 2973 3541 7043 12712 15005 17149 19910 36795 22nd row: 10091267 1380 4919 12679 22889 29638 30987 34637 36232 37284 23rd row: 466913 1247 1646 3049 5924 9014 20539 34546 35029 36540 24th row: 374 697984 1654 5870 10883 11684 20294 28888 31612 34031 25th row: 117 240 6355093 8673 11323 12456 14145 21397 39619 42559 26th row: 122 1265 142713528 14282 15241 16852 17227 34723 36836 39791 27th row: 595 1180 13106952 17916 24725 24971 27243 29555 32138 35987 28th row: 140 470 101713222 13253 18462 20806 21117 28673 31598 37235 29th row: 7 710 10728014 10804 13303 14292 16690 26676 36443 41966 30th row: 48 189 75912438 14523 16388 23178 27315 28656 29111 29694 31st row: 285 387 4104294 4467 5949 25386 27898 34880 41169 42614 32nd row: 474 545 132010506 13186 18126 27110 31498 35353 36193 37322 33rd row: 1075 1130 142411390 13312 14161 16927 25071 25844 34287 38151 34th row: 161 396 4275944 17281 22201 25218 30143 35566 38261 42513 35th row: 233 247 6941446 3180 3507 9069 20764 21940 33422 39358 36th row: 271 508 1013 627121760 21858 24887 29808 31099 35475 39924 37th row: 8 674 1329 3135 511014460 28108 28388 31043 31137 31863 38th row: 1035 1222 1409 8287 1608324450 24888 29356 30329 37834 39684 39th row: 391 1090 1128 1866 409510643 13121 14499 20056 22195 30593 40th row: 55 161 1402 6289 6837 879117937 21425 26602 30461 37241 41st row: 110 377 1228 6875 13253 1703219008 23274 32285 33452 41630 42nd row: 360 638 1355 5933 12593 1353323377 23881 24586 26040 41663 43rd row: 535 1240 1333 3354 10860 1603232573 34908 34957 39255 40759 44th row: 526 936 1321 7992 10260 1852728248 29356 32636 34666 35552 45th row: 336 785 875 7530 13062 1307518925 27963 28703 33688 36502 46th row: 36 591 1062 1518 3821 7048 1119717781 19408 22731 24783 47th row: 214 1145 1223 1546 9475 11170 1606121273 38688 40051 42479 48th row: 1136 1226 1423 20227 22573 24951 2646229586 34915 42441 43048 49th row: 26 276 1425 6048 7224 7917 8747 2755928515 35002 37649 50th row: 127 294 437 4029 8585 9647 11904 24115 2851436893 39722 51st row: 748 1093 1403 9536 19305 20468 31049 38667 4050240720 41949 52nd row: 96 638 743 9806 12101 17751 22732 24937 3200732594 38504 53rd row: 649 904 1079 2770 3337 9158 20125 24619 3292133698 35173 54th row: 401 518 984 7372 12438 12582 18704 35874 3942039503 39790 55th row: 10 451 1077 8078 16320 17409 25807 28814 3061341261 42955 56th row: 405 592 1178 15936 18418 19585 21966 24219 3063734536 37838 57th row: 50 584 851 9720 11919 22544 22545 25851 3556741587 41876 58th row: 911 1113 1176 1806 10058 10809 14220 19044 2074829424 36671 59th row: 441 550 1135 1956 11254 18699 30249 33099 3458735243 39952 60th row: 510 1016 1281 8621 13467 13780 15170 16289 2092526426 34479 61st row: 4969 5223 17117 21950 22144 24043 27151 39809 62ndrow: 11452 13622 18918 19670 23995 32647 37200 37399 63rd row: 6351 642613185 13973 16699 22524 31070 31916 64th row: 4098 10617 14854 1800428580 36158 37500 38552

The accumulation may be performed while the rows of the sequence arebeing repeatedly changed by the CPM size of the PCM.

In accordance with an aspect of the present invention, there is providedan LDPC encoding method, including initializing first memory configuredto store an LDPC codeword and second memory; and generating the LDPCcodeword corresponding to information bits by performing accumulationwith respect to the second memory using a sequence corresponding to aPCM.

The accumulation may be performed at parity bit addresses that areupdated using the sequence corresponding to the PCM.

The LDPC codeword may include a systematic part corresponding to theinformation bits and having a length of 21600, a first parity partcorresponding to a dual diagonal matrix included in the PCM and having alength of 1440, and a second parity part corresponding to an identitymatrix included in the PCM and having a length of 41760.

The sequence may have a number of rows equal to the sum of a valueobtained by dividing a length of the systematic part, that is, 21600, bya circulant permutation matrix (CPM) size corresponding to the PCM, thatis, 360, and a value obtained by dividing a length of the first paritypart, that is, 1440, by the CPM size.

The sequence may be represented by the above Sequence Table.

In accordance with still another aspect of the present invention, thereis provided an LDPC decoder, including a receiving unit configured toreceive an LDPC codeword encoded using a sequence corresponding to a PCMand is represented by the above Sequence Table; and a decoding unitconfigured to restore information bits from the received LDPC codewordby performing decoding corresponding to the PCM.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other objects, features and advantages of the presentinvention will be more clearly understood from the following detaileddescription taken in conjunction with the accompanying drawings, inwhich:

FIG. 1 is a block diagram illustrating a broadcast signal transmissionand reception system according to an embodiment of the presentinvention;

FIG. 2 is an operation flowchart illustrating a broadcast signaltransmission and reception method according to an embodiment of thepresent invention;

FIG. 3 is a diagram illustrating the structure of a PCM corresponding toan LDPC code to according to an embodiment of the present invention;

FIG. 4 is a block diagram illustrating an LDPC encoder according to anembodiment of the present invention;

FIG. 5 is a block diagram illustrating an LDPC decoder according to anembodiment of the present invention;

FIG. 6 is an operation flowchart illustrating an LDPC encoding methodaccording to an embodiment of the present invention; and

FIG. 7 is a graph plotting the performance of a QC-LDPC code having alength of 64800 and a code rate of 5/15 according to an embodiment ofthe present invention against E_(b)/N_(o).

DETAILED DESCRIPTION

Embodiments of the present invention will be described in detail belowwith reference to the accompanying drawings. Repeated descriptions anddescriptions of well-known functions and configurations that have beendeemed to make the gist of the present invention unnecessarily obscurewill be omitted below. The embodiments of the present invention areintended to fully describe the present invention to persons havingordinary knowledge in the art to which the present invention pertains.Accordingly, the shapes, sizes, etc. of components in the drawings maybe exaggerated to make the description obvious.

Embodiments of the present invention will be described in detail belowwith reference to the accompanying drawings.

FIG. 1 is a block diagram illustrating a broadcast signal transmissionand reception system according to an embodiment of the presentinvention.

Referring to FIG. 1, it can be seen that a transmitter 10 and a receiver30 communicate with each other over a wireless channel 20.

The transmitter 10 generates an n-bit codeword by encoding k informationbits using an LDPC encoder 13. The codeword is modulated by themodulator 15, and is transmitted via an antenna 17. The signaltransmitted via the wireless channel 20 is received via the antenna 31of the receiver 30, and, in the receiver 30, is subjected to a processreverse to the process in the transmitter 10. That is, the received datais demodulated by a demodulator 33, and is then decoded by an LDPCdecoder 35, thereby finally restoring the information bits.

It will be apparent to those skilled in the art that the above-describedtransmission and reception processes have been described within aminimum range required for a description of the features of the presentinvention and various processes required for data transmission may beadded.

In the following, the specific processes of encoding and decoding thatare performed using an LDPC code in the LDPC encoder 13 or LDPC decoder35 and the specific configurations of encoding and decoding devices,such as the LDPC encoder 13 and the LDPC decoder 35, are described. TheLDPC encoder 13 illustrated in FIG. 1 may have a structure illustratedin FIG. 4, and the LDPC decoder 35 may have a structure illustrated inFIG. 5.

FIG. 2 is an operation flowchart illustrating a broadcast signaltransmission and reception method according to an embodiment of thepresent invention.

Referring to FIG. 2, in the broadcast signal transmission and receptionmethod according to this embodiment of the present invention, input bits(information bits) are subjected to LDPC encoding at step S210.

That is, at step S210, an n-bit codeword is generated by encoding kinformation bits using the LDPC encoder.

In this case, step S210 may be performed as in an LDPC encoding methodillustrated in FIG. 6.

Furthermore, in the broadcast signal transmission and reception method,the encoded data is modulated at step S220.

That is, at step S220, the encoded n-bit codeword is modulated using themodulator.

Furthermore, in the broadcast signal transmission and reception method,the modulated data is transmitted at step S230.

That is, at step S230, the modulated codeword is transmitted over awireless channel via the antenna.

Furthermore, in the broadcast signal transmission and reception method,the received data is demodulated at step S240.

That is, at step S240, the signal transmitted over the wireless channelis received via the antenna of the receiver, and the received data isdemodulated using the demodulator.

Furthermore, in the broadcast signal transmission and reception method,the demodulated data is subjected to LDPC decoding at step S250.

That is, at step S250, the information bits are finally restored byperforming LDPC decoding using the demodulator of the receiver.

In this case, step S250 corresponds to a process reverse to that of theLDPC encoding method illustrated in FIG. 6, and may correspond to theLDPC decoder of FIG. 5.

An LDPC code is known as a code very close to the Shannon limit for anadditive white Gaussian noise (AWGN) channel, and has the advantages ofasymptotically excellent performance and parallelizable decodingcompared to a turbo code.

Generally, an LDPC code is defined by a low-density parity check matrix(PCM) that is randomly generated. However, a randomly generated LDPCcode requires a large amount of memory to store a PCM, and requires alot of time to access memory. In order to overcome these problems, aquasi-cyclic LDPC (QC-LDPC) code has been proposed. A QC-LDPC code thatis composed of a zero matrix or a circulant permutation matrix (CPM) isdefined by a PCM that is expressed by the following Equation 1:

$\begin{matrix}{{H = \begin{bmatrix}J^{a_{11}} & J^{a_{12}} & \ldots & J^{a_{1n}} \\J^{a_{21}} & J^{a_{22}} & \ldots & J^{a_{2n}} \\\vdots & \vdots & \ddots & \vdots \\J^{a_{m\; 1}} & J^{a_{m\; 2}} & \ldots & J^{a_{mn}}\end{bmatrix}},{{{for}\mspace{14mu} a_{ij}} \in \left\{ {0,1,\ldots\mspace{14mu},{L - 1},\infty} \right)}} & (1)\end{matrix}$

In this equation, J is a CPM having a size of L×L, and is given as thefollowing Equation 2. In the following description, L may be 360.

$\begin{matrix}{J_{L \times L} = \begin{bmatrix}0 & 1 & 0 & \ldots & 0 \\0 & 0 & 1 & \ldots & 0 \\\vdots & \vdots & \vdots & \ddots & \vdots \\0 & 0 & 0 & \ldots & 1 \\1 & 0 & 0 & \ldots & 0\end{bmatrix}} & (2)\end{matrix}$Furthermore, is obtained by shifting an L×L identity matrix I (J⁰) tothe right i (0≤i<L) times, and J^(∞) is an L×L zero matrix. Accordingly,in the case of a QC-LDPC code, it is sufficient if only index exponent iis stored in order to store J^(i), and thus the amount of memoryrequired to store a PCM is considerably reduced.

FIG. 3 is a diagram illustrating the structure of a PCM corresponding toan LDPC code to according to an embodiment of the present invention.

Referring to FIG. 3, the sizes of matrices A and C are g×K and(N−K−g)×(K+g), respectively, and are composed of an L×L zero matrix anda CPM, respectively. Furthermore, matrix Z is a zero matrix having asize of g×(N−K−g), matrix D is an identity matrix having a size of(N−K−g)×(N−K−g), and matrix B is a dual diagonal matrix having a size ofg×g. In this case, the matrix B may be a matrix in which all elementsexcept elements along a diagonal line and neighboring elements below thediagonal line are 0, and may be defined as the following Equation 3:

$\begin{matrix}{B_{g \times g} = \begin{bmatrix}I_{L \times L} & 0 & 0 & \ldots & 0 & 0 & 0 \\I_{L \times L} & I_{L \times L} & 0 & \ldots & 0 & 0 & 0 \\0 & I_{L \times L} & I_{L \times L} & \vdots & 0 & 0 & 0 \\\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\0 & 0 & 0 & \ldots & I_{L \times L} & I_{L \times L} & 0 \\0 & 0 & 0 & \ldots & 0 & I_{L \times L} & I_{L \times L}\end{bmatrix}} & (3)\end{matrix}$where I_(L×L) is an identity matrix having a size of L×L.

That is, the matrix B may be a bit-wise dual diagonal matrix, or may bea block-wise dual diagonal matrix having identity matrices as itsblocks, as indicated by Equation 3. The bit-wise dual diagonal matrix isdisclosed in detail in Korean Patent Application Publication No.2007-0058438, etc.

In particular, it will be apparent to those skilled in the art that whenthe matrix B is a bit-wise dual diagonal matrix, it is possible toperform conversion into a Quasi-cyclic form by applying row or columnpermutation to a PCM including the matrix B and having a structureillustrated in FIG. 3.

In this case, N is the length of a codeword, and K is the length ofinformation.

The present invention proposes a newly designed QC-LDPC code in whichthe code rate thereof is 5/15 and the length of a codeword is 64800, asillustrated in the following Table 1. That is, the present inventionproposes an LDPC code that is designed to receive information having alength of 21600 and generate an LDPC codeword having a length of 64800.

Table 1 illustrates the sizes of the matrices A, B, C, D and Z of theQC-LDPC code according to the present invention:

TABLE 1 Sizes Code rate Length A B C D Z 5/15 64800 1440 × 21600 1440 ×1440 41760 × 23040 41760 × 41760 1080 × 41760

The newly designed LDPC code may be represented in the form of asequence (progression), an equivalent relationship is establishedbetween the sequence and matrix (parity bit check matrix), and thesequence may be represented, as follows:

Sequence Table 1st row: 221 1011 1218 4299 7143 8728 11072 15533 1735633909 36833 2nd row: 360 1210 1375 2313 3493 16822 21373 23588 2365626267 34098 3rd row: 544 1347 1433 2457 9186 10945 13583 14858 1919534606 37441 4th row: 37 596 715 4134 8091 12106 24307 24658 34108 4059142883 5th row: 235 398 1204 2075 6742 11670 13512 23231 24784 2791534752 6th row: 204 873 890 13550 16570 19774 34012 35249 37655 3988542890 7th row: 221 371 514 11984 14972 15690 28827 29069 30531 3101843121 8th row: 280 549 1435 1889 3310 10234 11575 15243 20748 3046936005 9th row: 223 666 1248 13304 14433 14732 18943 21248 23127 3852939272 10th row: 370 819 1065 9461 10319 25294 31958 33542 37458 3968140039 11st row: 585 870 1028 5087 5216 12228 16216 16381 16937 2713227893 12nd row: 164 167 1210 7386 11151 20413 22713 23134 24188 3677138992 13rd row: 298 511 809 4620 7347 8873 19602 24162 29198 34304 4114514th row: 105 830 1212 2415 14759 15440 16361 16748 22123 32684 4257515th row: 659 665 668 6458 22130 25972 30697 31074 32048 36078 3712916th row: 91 808 953 8015 8988 13492 13987 15979 28355 34509 39698 17throw: 594 983 1265 3028 4029 9366 11069 11512 27066 40939 41639 18th row:506 740 1321 1484 10747 16376 17384 20285 31502 38925 42606 19th row:338 356 975 2022 3578 18689 18772 19826 22914 24733 27431 20th row: 7091264 1366 4617 8893 25226 27800 29080 30277 37781 39644 21st row: 8401179 1338 2973 3541 7043 12712 15005 17149 19910 36795 22nd row: 10091267 1380 4919 12679 22889 29638 30987 34637 36232 37284 23rd row: 466913 1247 1646 3049 5924 9014 20539 34546 35029 36540 24th row: 374 697984 1654 5870 10883 11684 20294 28888 31612 34031 25th row: 117 240 6355093 8673 11323 12456 14145 21397 39619 42559 26th row: 122 1265 142713528 14282 15241 16852 17227 34723 36836 39791 27th row: 595 1180 13106952 17916 24725 24971 27243 29555 32138 35987 28th row: 140 470 101713222 13253 18462 20806 21117 28673 31598 37235 29th row: 7 710 10728014 10804 13303 14292 16690 26676 36443 41966 30th row: 48 189 75912438 14523 16388 23178 27315 28656 29111 29694 31st row: 285 387 4104294 4467 5949 25386 27898 34880 41169 42614 32nd row: 474 545 132010506 13186 18126 27110 31498 35353 36193 37322 33rd row: 1075 1130 142411390 13312 14161 16927 25071 25844 34287 38151 34th row: 161 396 4275944 17281 22201 25218 30143 35566 38261 42513 35th row: 233 247 6941446 3180 3507 9069 20764 21940 33422 39358 36th row: 271 508 1013 627121760 21858 24887 29808 31099 35475 39924 37th row: 8 674 1329 3135 511014460 28108 28388 31043 31137 31863 38th row: 1035 1222 1409 8287 1608324450 24888 29356 30329 37834 39684 39th row: 391 1090 1128 1866 409510643 13121 14499 20056 22195 30593 40th row: 55 161 1402 6289 6837 879117937 21425 26602 30461 37241 41st row: 110 377 1228 6875 13253 1703219008 23274 32285 33452 41630 42nd row: 360 638 1355 5933 12593 1353323377 23881 24586 26040 41663 43rd row: 535 1240 1333 3354 10860 1603232573 34908 34957 39255 40759 44th row: 526 936 1321 7992 10260 1852728248 29356 32636 34666 35552 45th row: 336 785 875 7530 13062 1307518925 27963 28703 33688 36502 46th row: 36 591 1062 1518 3821 7048 1119717781 19408 22731 24783 47th row: 214 1145 1223 1546 9475 11170 1606121273 38688 40051 42479 48th row: 1136 1226 1423 20227 22573 24951 2646229586 34915 42441 43048 49th row: 26 276 1425 6048 7224 7917 8747 2755928515 35002 37649 50th row: 127 294 437 4029 8585 9647 11904 24115 2851436893 39722 51st row: 748 1093 1403 9536 19305 20468 31049 38667 4050240720 41949 52nd row: 96 638 743 9806 12101 17751 22732 24937 3200732594 38504 53rd row: 649 904 1079 2770 3337 9158 20125 24619 3292133698 35173 54th row: 401 518 984 7372 12438 12582 18704 35874 3942039503 39790 55th row: 10 451 1077 8078 16320 17409 25807 28814 3061341261 42955 56th row: 405 592 1178 15936 18418 19585 21966 24219 3063734536 37838 57th row: 50 584 851 9720 11919 22544 22545 25851 3556741587 41876 58th row: 911 1113 1176 1806 10058 10809 14220 19044 2074829424 36671 59th row: 441 550 1135 1956 11254 18699 30249 33099 3458735243 39952 60th row: 510 1016 1281 8621 13467 13780 15170 16289 2092526426 34479 61st row: 4969 5223 17117 21950 22144 24043 27151 39809 62ndrow: 11452 13622 18918 19670 23995 32647 37200 37399 63rd row: 6351 642613185 13973 16699 22524 31070 31916 64th row: 4098 10617 14854 1800428580 36158 37500 38552

An LDPC code that is represented in the form of a sequence is beingwidely used in the DVB standard.

According to an embodiment of the present invention, an LDPC codepresented in the form of a sequence is encoded, as follows. It isassumed that there is an information block S=(s₀, s₁, . . . , s_(K−1))having an information size K. The LDPC encoder generates a codewordΛ=(λ₀, λ₁, λ₂, . . . , λ_(N−1)) having a size of N=K+M₁+M₂ using theinformation block S having a size K. In this case, M₁=g, and M₂=N−K−g.Furthermore, M₁ is the size of parity bits corresponding to the dualdiagonal matrix B, and M₂ is the size of parity bits corresponding tothe identity matrix D. The encoding process is performed, as follows:

Initialization:λ_(i) =s _(i) for i=0,1, . . . ,K−1p _(j)=0 for j=0.1, . . . ,M ₂−1  (4)

First information bit λ₀ is accumulated at parity bit addressesspecified in the 1st row of the sequence of the Sequence Table. Forexample, in an LDPC code having a length of 64800 and a code rate of5/15, an accumulation process is as follows:

p₂₂₁=p₂₂₁⊕λ₀ p₁₀₁₁=p₁₀₁₁⊕λ₀ p₁₂₁₈=p₁₂₁₈⊕λ₀ p₄₂₉₉=p₄₂₉₉⊕λ₀ p₇₁₄₃=p₇₁₄₃⊕λ₀

p₈₇₂₈=p₈₇₂₈⊕λ₀ p₁₁₀₇₂=p₁₁₀₇₂⊕λ₀ p₁₅₅₃₃=p₁₅₅₃₃⊕λ₀ p₁₇₃₅₆=p₁₇₃₅₆⊕λ₀

p₃₃₉₀₉=p₃₃₉₀₉⊕λ₀ p₃₆₈₃₃=p₃₆₈₃₃⊕λ₀

where the addition ⊕ occurs in GF(2).

The subsequent L−1 information bits, that is, λ_(m), m=1, 2, . . . ,L−1, are accumulated at parity bit addresses that are calculated by thefollowing Equation 5:(x+M×Q ₁)mod M ₁ if x<M ₁M ₁+{(x−M ₁ ×Q ₂)mod M ₂} if x≥M ₁  (5)where x denotes the addresses of parity bits corresponding to the firstinformation bit λ₀, that is, the addresses of the parity bits specifiedin the first row of the sequence of the Sequence Table, Q₁=M₁/L,Q₂=M₂/L, and L=360. Furthermore, Q₁ and Q₂ are defined in the followingTable 2. For example, for an LDPC code having a length of 64800 and acode rate of 5/15, M₁=1440, Q₁=4, M₂=41760, Q₂=116 and L=360, and thefollowing operations are performed on the second bit λ₁ using Equation5:p₂₂₅=p₂₂₅⊕λ₁ p₁₀₁₅=p₁₀₁₅⊕λ₁ p₁₂₂₂=p₁₂₂₂⊕λ₁ p₄₄₁₅=p₄₄₁₅Γλ₁ p₇₂₅₉=p₇₂₅₉⊕λ₁p₈₈₄₄=p₈₈₄₄⊕λ₁ p₁₁₁₈₈=p₁₁₁₈₈⊕λ₁ p₁₅₆₄₉=p₁₅₆₄₉⊕λ₁ p₁₇₄₇₂=p₁₇₄₇₂⊕λ₁p₃₄₀₂₅=p₃₄₀₂₅⊕λ₁ p₃₆₉₄₉=p₃₆₉₄₉⊕λ₁

Table 2 illustrates the sizes of M₁, Q₁, M₂ and Q₂ of the designedQC-LDPC code:

TABLE 2 Sizes Code rate Length M₁ M₂ Q₁ Q₂ 5/15 64800 1440 41760 4 116

The addresses of parity bit accumulators for new 360 information bitsfrom λ_(L) to λ_(2L−1) are calculated and accumulated from Equation 5using the second row of the sequence.

In a similar manner, for all groups composed of new L information bits,the addresses of parity bit accumulators are calculated and accumulatedfrom Equation 5 using new rows of the sequence.

After all the information bits from λ₀ to λ_(K−1) have been exhausted,the operations of the following Equation 6 are sequentially performedfrom i=1:p _(i) =p _(i) ⊕p _(i−1) for i=0,1, . . . ,M ₁−1  (6)

Thereafter, when a parity interleaving operation, such as that of thefollowing Equation 7, is performed, parity bits corresponding to thedual diagonal matrix B are generated:λ_(K+L·t+s) =p _(Q) ₁ _(·s+t) for 0≤s<L,0≤t<Q ₁  (7)

When the parity bits corresponding to the dual diagonal matrix B havebeen generated using K information bits λ₀, λ₁, . . . , λ_(K−1), paritybits corresponding to the identity matrix D are generated using the M₁generated parity bits λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹.

For all groups composed of L information bits from λ_(K) to λ_(K+M) ₁⁻¹, the addresses of parity bit accumulators are calculated using thenew rows (starting with a row immediately subsequent to the last rowused when the parity bits corresponding to the dual diagonal matrix Bhave been generated) of the sequence and Equation 5, and relatedoperations are performed.

When a parity interleaving operation, such as that of the followingEquation 8, is performed after all the information bits from λ_(K) toλ_(K+M) ₁ ⁻¹ have been exhausted, parity bits corresponding to theidentity matrix D are generated:λ_(K+M) ₁ _(+L·t+s) =p _(M) ₁ _(+Q) ₂ _(·s+t) for 0≤s<L,0≤t<Q ₂  (8)

FIG. 4 is a block diagram illustrating an LDPC encoder according to anembodiment of the present invention.

Referring to FIG. 4, the LDPC encoder according to this embodiment ofthe present invention includes memory 310 and 320 and a processor 330.

The memory 310 is memory that is used to store an LDPC codeword having alength of 64800 and a code rate of 5/15.

The memory 320 is memory that is initialized to 0.

The memory 310 and the memory 320 may correspond to λ_(i) (1=0, 1, . . ., N−1) and p_(j) (j=0, 1, . . . , M₁+M₂−1), respectively.

The memory 310 and the memory 320 may correspond to various types ofhardware for storing sets of bits, and may correspond to datastructures, such as an array, a list, a stack and a queue.

The processor 330 generates an LDPC codeword corresponding toinformation bits by performing accumulation with respect to the memory320 using a sequence corresponding to a PCM.

In this case, the accumulation may be performed at parity bit addressesthat are updated using the sequence of the above Sequence Table.

In this case, the LDPC codeword may include a systematic part λ₀, λ₁, .. . , λ_(K−1) corresponding to the information bits and having a lengthof 21600 (=K), a first parity part λ_(K), λ_(K+1), λ_(K+M) ₁ ⁻¹corresponding to a dual diagonal matrix included in the PCM and having alength of 1440 (=M₁=g), and a second parity part λ_(K+M) ₁ , λ_(K+M) ₁₊₁, . . . , λ_(K+M) ₁ _(+M) ₂ ⁻¹ corresponding to an identity matrixincluded in the PCM and having a length of 41760 (=M₂).

In this case, the sequence may have a number of rows equal to the sum(21600/360+1440/360=64) of a value obtained by dividing the length ofthe systematic part, that is, 21600, by a CPM size L corresponding tothe PCM, that is, 360, and a value obtained by dividing the length M₁ ofthe first parity part, that is, 1440, by 360.

As described above, the sequence may be represented by the aboveSequence Table.

In this case, the memory 320 may have a size corresponding to the sumM₁+M₂ of the length M₁ of the first parity part and the length M₂ of thesecond parity part.

In this case, the parity bit addresses may be updated based on theresults of comparing each x of the previous parity bit addressesspecified in respective rows of the sequence with the length M₁ of thefirst parity part.

That is, the parity bit addresses may be updated using Equation 5. Inthis case, x may be the previous parity bit addresses, m may be aninformation bit index that is an integer larger than 0 and smaller thanL, L may be the CPM size of the PCM, Q₁ may be M₁/L, M₁ may be the sizeof the first parity part, Q₂ may be M₂/L, and M₂ may be the size of thesecond parity part.

In this case, it may be possible to perform the accumulation whilerepeatedly changing the rows of the sequence by the CPM size L (=360) ofthe PCM, as described above.

In this case, the first parity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹may be generated by performing parity interleaving using the memory 310and the memory 320, as described in conjunction with Equation 7.

In this case, the second parity part λ_(K+M) ₁ , λ_(K+M) ₁ ⁺¹, . . . ,λ_(K+M) ₁ _(+M) ₂ ⁻¹ may be generated by performing parity interleavingusing the memory 310 and the memory 320 after generating the firstparity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹ and then performing theaccumulation using the first parity part λ_(K), λ_(K+1), . . . , λ_(K+M)₁ ⁻¹ and the sequence, as described in conjunction with Equation 8.

FIG. 5 is a block diagram illustrating an LDPC decoder according to anembodiment of the present invention.

Referring to FIG. 5, the LDPC decoder according to this embodiment ofthe present invention may include a receiving unit 410 and a decodingunit 420.

The receiving unit 410 receives an LDPC codeword that has been encodedusing a sequence that corresponds to a PCM and is represented by theabove Sequence Table.

The decoding unit 420 restores information bits from the received LDPCcodeword by performing decoding corresponding to the PCM.

In this case, the sequence may be used to update the parity bitaddresses of the memory, and the parity bit addresses are used foraccumulation that is performed to generate parity bits corresponding tothe LDPC codeword.

In this case, the LDPC codeword may include a systematic part λ₀, λ₁, .. . , λ_(K−1) corresponding to the information bits, a first parity partλ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹ corresponding to a dual diagonalmatrix included in the PCM, and a second parity part λ_(K+M) ₁ , λ_(K+M)₁ ₊₁, . . . , λ_(K+M) ₁ _(+M) ₂ ⁻¹ corresponding to an identity matrixincluded in the PCM.

In this case, the parity bit addresses may be updated based on theresults of comparing each x of the previous parity bit addressesspecified in respective rows of the sequence with the length M₁ of thefirst parity part.

That is, the parity bit addresses may be updated using Equation 5. Inthis case, x may be the previous parity bit addresses, m may be aninformation bit index that is an integer larger than 0 and smaller thanL, L may be the CPM size of the PCM, Q₁ may be M₁/L, M₁ may be the sizeof the first parity part, Q₂ may be M₂/L, and M₂ may be the size of thesecond parity part.

FIG. 6 is an operation flowchart illustrating an LDPC encoding methodaccording to an embodiment of the present invention.

Referring to FIG. 6, the LDPC encoding method according to thisembodiment of the present invention initializes the first and secondmemory that stores an LDPC codeword at step S510.

In this case, step S510 may be performed using Equation 4.

Furthermore, in the LDPC encoding method according to this embodiment ofthe present invention, an LDPC codeword corresponding to informationbits is generated by performing accumulation with respect to the secondmemory using a sequence corresponding to a PCM at step S520.

In this case, the accumulation may be performed at parity bit addressesthat are updated using the sequence corresponding to the PCM.

In this case, the LDPC codeword may include a systematic part λ₀, λ₁, .. . , λ_(K−1) corresponding to the information bits and having a lengthof 21600 (=K), a first parity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹corresponding to a dual diagonal matrix included in the PCM and having alength of 1440 (=M₁=_(g)), and a second parity part λ_(K+M) ₁ , λ_(K+M)₁ ₊₁, . . . , λ_(K+M) ₁ _(+M) ₂ ⁻¹ corresponding to an identity matrixincluded in the PCM and having a length of 41760 (=M₂).

In this case, the sequence may have a number of rows equal to the sum(21600/360+1440/360=64) of a value obtained by dividing the length ofthe systematic part, that is, 21600, by a CPM size L corresponding tothe PCM, that is, 360, and a value obtained by dividing the length M₁ ofthe first parity part, that is, 1440, by 360.

As described above, the sequence may be represented by the aboveSequence Table.

In this case, the parity bit addresses may be updated based on theresults of comparing each x of the previous parity bit addressesspecified in respective rows of the sequence with the length M₁ of thefirst parity part.

That is, the parity bit addresses may be updated using Equation 5. Inthis case, x may be the previous parity bit addresses, m may be aninformation bit index that is an integer larger than 0 and smaller thanL, L may be the CPM size of the PCM, Q₁ may be M₁/L, M₁ may be the sizeof the first parity part, Q₂ may be M₂/L, and M₂ may be the size of thesecond parity part.

In this case, it may be possible to perform the accumulation whilerepeatedly changing the rows of the sequence by the CPM size L (=360) ofthe PCM, as described above.

In this case, the first parity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹may be generated by performing parity interleaving using the memory 310and the memory 320, as described in conjunction with Equation 7.

In this case, the second parity part λ_(K+M) ₁ , λ_(K+M) ₁ ₊₁, . . . ,λ_(K+M) ₁ _(+M) ₂ ⁻¹ may be generated by performing parity interleavingusing the memory 310 and the memory 320 after generating the firstparity part λ_(K), λ_(K+1), . . . , λ_(K+M) ₁ ⁻¹ and then performing theaccumulation using the first parity part λ_(K), λ_(K+1), . . . , λ_(K+M)₁ ⁻¹ and the sequence, as described in conjunction with Equation 8.

FIG. 7 is a graph plotting the performance of a QC-LDPC code having alength of 64800 and a code rate of 5/15 according to an embodiment ofthe present invention against E_(b)/N_(o).

The graph illustrated in FIG. 7 illustrates results that were obtainedon the assumption that a log-likelihood ratio (LLR)-based sum-productalgorithm in which binary phase shift keying (BPSK) modulation and 50rounds of repetitive decoding were performed was used for computationalexperiments. As illustrated in FIG. 7, it can be seen that the designedcode is away from the Shannon limit by about 0.55 dB at BER=10⁻⁶.

At least one embodiment of the present invention has the advantage ofproviding a new LDPC codeword having a length of 64800 and a code rateof 5/15, which is capable of being used for general purposes.

At least one embodiment of the present invention has the advantage ofproviding an LDPC encoding technique that is capable of efficientlyperforming LDPC encoding using a sequence having a number of rows equalto a value that is obtained by dividing the sum of the length of thesystematic part of an LDPC codeword, that is, 21600, and the length ofthe first parity part of the LDPC codeword, that is, 1440, by 360.

Although the specific embodiments of the present invention have beendisclosed for illustrative purposes, those skilled in the art willappreciate that various modifications, additions and substitutions arepossible without departing from the scope and spirit of the invention asdisclosed in the accompanying claims.

What is claimed is:
 1. A low density parity check (LDPC) decoder,comprising: a receiving unit configured to receive a signalcorresponding to an LDPC codeword having a length of 64800 and a coderate of 5/15, the LDPC codeword encoded using a sequence correspondingto a parity check matrix (PCM); and a decoding unit configured toperform decoding the received signal, the decoding corresponding to theparity check matrix, wherein the LDPC codeword comprises a systematicpart corresponding to the information bits and having a length of 21600,a first parity part corresponding to a dual diagonal matrix included inthe PCM and having a length of 1440, and a second parity partcorresponding to an identity matrix included in the PCM and having alength of 41760, wherein the LDPC codeword is generated by performingaccumulation with respect to memory using the sequence, and theaccumulation is performed at parity bit addresses that are updated basedon results of comparing each of previous parity bit addresses specifiedin each row of the sequence with the size of the first parity part, andwherein the sequence is represented by the following Sequence Table:Sequence Table 1st row: 221 1011 1218 4299 7143 8728 11072 15533 1735633909 36833 2nd row: 360 1210 1375 2313 3493 16822 21373 23588 2365626267 34098 3rd row: 544 1347 1433 2457 9186 10945 13583 14858 1919534606 37441 4th row: 37 596 715 4134 8091 12106 24307 24658 34108 4059142883 5th row: 235 398 1204 2075 6742 11670 13512 23231 24784 2791534752 6th row: 204 873 890 13550 16570 19774 34012 35249 37655 3988542890 7th row: 221 371 514 11984 14972 15690 28827 29069 30531 3101843121 8th row: 280 549 1435 1889 3310 10234 11575 15243 20748 3046936005 9th row: 223 666 1248 13304 14433 14732 18943 21248 23127 3852939272 10th row: 370 819 1065 9461 10319 25294 31958 33542 37458 3968140039 11st row: 585 870 1028 5087 5216 12228 16216 16381 16937 2713227893 12nd row: 164 167 1210 7386 11151 20413 22713 23134 24188 3677138992 13rd row: 298 511 809 4620 7347 8873 19602 24162 29198 34304 4114514th row: 105 830 1212 2415 14759 15440 16361 16748 22123 32684 4257515th row: 659 665 668 6458 22130 25972 30697 31074 32048 36078 3712916th row: 91 808 953 8015 8988 13492 13987 15979 28355 34509 39698 17throw: 594 983 1265 3028 4029 9366 11069 11512 27066 40939 41639 18th row:506 740 1321 1484 10747 16376 17384 20285 31502 38925 42606 19th row:338 356 975 2022 3578 18689 18772 19826 22914 24733 27431 20th row: 7091264 1366 4617 8893 25226 27800 29080 30277 37781 39644 21st row: 8401179 1338 2973 3541 7043 12712 15005 17149 19910 36795 22nd row: 10091267 1380 4919 12679 22889 29638 30987 34637 36232 37284 23rd row: 466913 1247 1646 3049 5924 9014 20539 34546 35029 36540 24th row: 374 697984 1654 5870 10883 11684 20294 28888 31612 34031 25th row: 117 240 6355093 8673 11323 12456 14145 21397 39619 42559 26th row: 122 1265 142713528 14282 15241 16852 17227 34723 36836 39791 27th row: 595 1180 13106952 17916 24725 24971 27243 29555 32138 35987 28th row: 140 470 101713222 13253 18462 20806 21117 28673 31598 37235 29th row: 7 710 10728014 10804 13303 14292 16690 26676 36443 41966 30th row: 48 189 75912438 14523 16388 23178 27315 28656 29111 29694 31st row: 285 387 4104294 4467 5949 25386 27898 34880 41169 42614 32nd row: 474 545 132010506 13186 18126 27110 31498 35353 36193 37322 33rd row: 1075 1130 142411390 13312 14161 16927 25071 25844 34287 38151 34th row: 161 396 4275944 17281 22201 25218 30143 35566 38261 42513 35th row: 233 247 6941446 3180 3507 9069 20764 21940 33422 39358 36th row: 271 508 1013 627121760 21858 24887 29808 31099 35475 39924 37th row: 8 674 1329 3135 511014460 28108 28388 31043 31137 31863 38th row: 1035 1222 1409 8287 1608324450 24888 29356 30329 37834 39684 39th row: 391 1090 1128 1866 409510643 13121 14499 20056 22195 30593 40th row: 55 161 1402 6289 6837 879117937 21425 26602 30461 37241 41st row: 110 377 1228 6875 13253 1703219008 23274 32285 33452 41630 42nd row: 360 638 1355 5933 12593 1353323377 23881 24586 26040 41663 43rd row: 535 1240 1333 3354 10860 1603232573 34908 34957 39255 40759 44th row: 526 936 1321 7992 10260 1852728248 29356 32636 34666 35552 45th row: 336 785 875 7530 13062 1307518925 27963 28703 33688 36502 46th row: 36 591 1062 1518 3821 7048 1119717781 19408 22731 24783 47th row: 214 1145 1223 1546 9475 11170 1606121273 38688 40051 42479 48th row: 1136 1226 1423 20227 22573 24951 2646229586 34915 42441 43048 49th row: 26 276 1425 6048 7224 7917 8747 2755928515 35002 37649 50th row: 127 294 437 4029 8585 9647 11904 24115 2851436893 39722 51st row: 748 1093 1403 9536 19305 20468 31049 38667 4050240720 41949 52nd row: 96 638 743 9806 12101 17751 22732 24937 3200732594 38504 53rd row: 649 904 1079 2770 3337 9158 20125 24619 3292133698 35173 54th row: 401 518 984 7372 12438 12582 18704 35874 3942039503 39790 55th row: 10 451 1077 8078 16320 17409 25807 28814 3061341261 42955 56th row: 405 592 1178 15936 18418 19585 21966 24219 3063734536 37838 57th row: 50 584 851 9720 11919 22544 22545 25851 3556741587 41876 58th row: 911 1113 1176 1806 10058 10809 14220 19044 2074829424 36671 59th row: 441 550 1135 1956 11254 18699 30249 33099 3458735243 39952 60th row: 510 1016 1281 8621 13467 13780 15170 16289 2092526426 34479 61st row: 4969 5223 17117 21950 22144 24043 27151 39809 62ndrow: 11452 13622 18918 19670 23995 32647 37200 37399 63rd row: 6351 642613185 13973 16699 22524 31070 31916 64th row: 4098 10617 14854 1800428580 36158 37500
 38552.


2. The LDPC decoder of claim 1, wherein the sequence has a number ofrows equal to a sum of a value obtained by dividing a length of thesystematic part by a circulant permutation matrix (CPM) sizecorresponding to the PCM and a value obtained by dividing a length ofthe first parity part by the CPM size.
 3. The LDPC decoder of claim 2,wherein the accumulation is performed while the rows of the sequence arebeing repeatedly changed by the CPM size of the PCM.
 4. A low densityparity check (LDPC) decoding method, comprising: receiving a signalcorresponding to an LDPC codeword having a length of 64800 and a coderate of 5/15, the LDPC codeword encoded using a sequence correspondingto a parity check matrix (PCM); and decoding the received signal, thedecoding corresponding to the parity check matrix, wherein the LDPCcodeword comprises a systematic part corresponding to the informationbits and having a length of 21600, a first parity part corresponding toa dual diagonal matrix included in the PCM and having a length of 1440,and a second parity part corresponding to an identity matrix included inthe PCM and having a length of 41760, wherein the LDPC codeword isgenerated by performing accumulation with respect to memory using thesequence, and the accumulation is performed at parity bit addresses thatare updated based on results of comparing each of previous parity bitaddresses specified in each row of the sequence with the size of thefirst parity part, and wherein the sequence is represented by thefollowing Sequence Table: Sequence Table  1st row: 221 1011 1218 42997143 8728 11072 15533 17356 33909 36833  2nd row: 360 1210 1375 23133493 16822 21373 23588 23656 26267 34098  3rd row: 544 1347 1433 24579186 10945 13583 14858 19195 34606 37441  4th row: 37 596 715 4134 809112106 24307 24658 34108 40591 42883  5th row: 235 398 1204 2075 674211670 13512 23231 24784 27915 34752  6th row: 204 873 890 13550 1657019774 34012 35249 37655 39885 42890  7th row: 221 371 514 11984 1497215690 28827 29069 30531 31018 43121  8th row: 280 549 1435 1889 331010234 11575 15243 20748 30469 36005  9th row: 223 666 1248 13304 1443314732 18943 21248 23127 38529 39272 10th row: 370 819 1065 9461 1031925294 31958 33542 37458 39681 40039 11th row: 585 870 1028 5087 521612228 16216 16381 16937 27132 27893 12nd row: 164 167 1210 7386 1115120413 22713 23134 24188 36771 38992 13rd row: 298 511 809 4620 7347 887319602 24162 29198 34304 41145 14th row: 105 830 1212 2415 14759 1544016361 16748 22123 32684 42575 15th row: 659 665 668 6458 22130 2597230697 31074 32048 36078 37129 16th row: 91 808 953 8015 8988 13492 1398715979 28355 34509 39698 17th row: 594 983 1265 3028 4029 9366 1106911512 27066 40939 41639 18th row: 506 740 1321 1484 10747 16376 1738420285 31502 38925 42606 19th row: 338 356 975 2022 3578 18689 1877219826 22914 24733 27431 20th row: 709 1264 1366 4617 8893 25226 2780029080 30277 37781 39644 21st row: 840 1179 1338 2973 3541 7043 1271215005 17149 19910 36795 22nd row: 1009 1267 1380 4919 12679 22889 2963830987 34637 36232 37284 23rd row: 466 913 1247 1646 3049 5924 9014 2053934546 35029 36540 24th row: 374 697 984 1654 5870 10883 11684 2029428888 31612 34031 25th row: 117 240 635 5093 8673 11323 12456 1414521397 39619 42559 26th row: 122 1265 1427 13528 14282 15241 16852 1722734723 36836 39791 27th row: 595 1180 1310 6952 17916 24725 24971 2724329555 32138 35987 28th row: 140 470 1017 13222 13253 18462 20806 2111728673 31598 37235 29th row: 7 710 1072 8014 10804 13303 14292 1669026676 36443 41966 30th row: 48 189 759 12438 14523 16388 23178 2731528656 29111 29694 31st row: 285 387 410 4294 4467 5949 25386 27898 3488041169 42614 32nd row: 474 545 1320 10506 13186 18126 27110 31498 3535336193 37322 33rd row: 1075 1130 1424 11390 13312 14161 16927 25071 2584434287 38151 34th row: 161 396 427 5944 17281 22201 25218 30143 3556638261 42513 35th row: 233 247 694 1446 3180 3507 9069 20764 21940 3342239358 36th row: 271 508 1013 6271 21760 21858 24887 29808 31099 3547539924 37th row: 8 674 1329 3135 5110 14460 28108 28388 31043 31137 3186338th row: 1035 1222 1409 8287 16083 24450 24888 29356 30329 37834 3968439th row: 391 1090 1128 1866 4095 10643 13121 14499 20056 22195 3059340th row: 55 161 1402 6289 6837 8791 17937 21425 26602 30461 37241 41strow: 110 377 1228 6875 13253 17032 19008 23274 32285 33452 41630 42ndrow: 360 638 1355 5933 12593 13533 23377 23881 24586 26040 41663 43rdrow: 535 1240 1333 3354 10860 16032 32573 34908 34957 39255 40759 44throw: 526 936 1321 7992 10260 18527 28248 29356 32636 34666 35552 45throw: 336 785 875 7530 13062 13075 18925 27963 28703 33688 36502 46throw: 36 591 1062 1518 3821 7048 11197 17781 19408 22731 24783 47th row:214 1145 1223 1546 9475 11170 16061 21273 38688 40051 42479 48th row:1136 1226 1423 20227 22573 24951 26462 29586 34915 42441 43048 49th row:26 276 1425 6048 7224 7917 8747 27559 28515 35002 37649 50th row: 127294 437 4029 8585 9647 11904 24115 28514 36893 39722 51st row: 748 10931403 9536 19305 20468 31049 38667 40502 40720 41949 52nd row: 96 638 7439806 12101 17751 22732 24937 32007 32594 38504 53rd row: 649 904 10792770 3337 9158 20125 24619 32921 33698 35173 54th row: 401 518 984 737212438 12582 18704 35874 39420 39503 39790 55th row: 10 451 1077 807816320 17409 25807 28814 30613 41261 42955 56th row: 405 592 1178 1593618418 19585 21966 24219 30637 34536 37838 57th row: 50 584 851 972011919 22544 22545 25851 35567 41587 41876 58th row: 911 1113 1176 180610058 10809 14220 19044 20748 29424 36671 59th row: 441 550 1135 195611254 18699 30249 33099 34587 35243 39952 60th row: 510 1016 1281 862113467 13780 15170 16289 20925 26426 34479 61st row: 4969 5223 1711721950 22144 24043 27151 39809 62nd row: 11452 13622 18918 19670 2399532647 37200 37399 63rd row: 6351 6426 13185 13973 16699 22524 3107031916 64th row: 4098 10617 14854 18004 28580 36158 37500 38552.